Tangles were originally introduced as a concept to formalize regions of high connectivity in graphs. In recent years, they have also been discovered as a link between structural graph theory and data science: when interpreting similarity in data sets as connectivity between points, finding clusters in the data essentially amounts to finding tangles in the underlying graphs. This paper explores the potential of tangles in data sets as a means for a formal quantitative study of clusters. Real-world data often follow a normal distribution. Accounting for this, we develop a quantitative theory of tangles in data sets drawn from Gaussian mixtures. To this end, we equip the data with a graph structure that models similarity between the points and allows us to apply tangle theory. We provide explicit conditions under which tangles associated with the marginal Gaussian distributions exist asymptotically almost surely. This can be considered a sufficient formal criterion for the separability of clusters in the data.